Does chess skill grow along a textbook learning curve?
Chess ratings of young players were better described by an exponential than a power learning curve, but many players broke both models by improving slowly at first and speeding up years later.
Source
Playing off the curve - testing quantitative predictions of skill acquisition theories in development of chess performance
Study at a glance
- Design
- Cohort — Archival longitudinal analysis of German chess federation ratings (1989 to 2007); power and exponential functions fitted to each player's yearly ratings and, separately, fitted to the first 5 years to predict later years
- N
- N=1383 · 1383 players who entered the national database between age 6 and 20 and played rated tournaments in each of at least 10 consecutive years, analysed in four starting-age groups
- Population
- German tournament chess players, predominantly male, who began competitive play in childhood or adolescence
- Outcome
- Individual fit and prediction error of power versus exponential learning functions for yearly chess ratings, year-to-year rating gains, and links with number of tournament games played
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What they did
The authors used the German chess federation's records, which track almost every rated game in the country, to follow 1383 players who began tournaments between ages 6 and 20 and kept playing for at least 10 years. For each player they fitted two classic learning functions, a power function and a negative exponential, to yearly ratings, and also fitted them to only the first 5 years to see which better predicted later ratings. They then looked at year-by-year gains, the number of games played, starting age and birth cohort.
What they found
The exponential function fitted the first years better for 88% of players and predicted later ratings better for 62%. However, both functions predict the biggest gains in the first year and shrinking gains afterwards, and many players, especially those starting young, showed the opposite: small or no gains early, then larger gains in later years. This pattern held across birth cohorts and was only partly explained by playing more tournament games over time. Early-year gains still predicted who would be strongest by year 10.
The limits
What it doesn't show
The data include only rated tournament games, not the amount or type of practice done outside tournaments, so the authors cannot tell whether the delayed improvement reflects changing practice, motivation or development. The sample is limited to German players who stayed in chess for at least a decade, which excludes those who quit and may not generalise to other countries or skills. The study is descriptive: it shows that standard learning curves misfit, and suggests a sigmoid pattern, but does not test a model that explains why.
Key terms
- Learning curve
- A graph or function describing how performance changes with practice or time.
- Power function vs exponential function
- Two shapes for learning curves; both show diminishing gains, but in an exponential curve the proportion of remaining improvement gained per unit of practice stays constant, whereas in a power curve it declines.
- Relative learning rate
- The share of the still-achievable improvement that is gained in each period; constant for exponential learning, decreasing for power-law learning.
- Elo rating
- A chess skill score calculated from results against opponents of known strength, used here as an interval measure of expertise.
- Prediction versus fit
- Fitting adjusts a model to existing data; prediction tests whether the fitted model forecasts new data, which is a stronger test because flexible models can fit almost anything.
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Quiz yourself
For most players, which function better fitted the first years of ratings?
Common questions
Why test prediction instead of just comparing fits?
A more flexible function can fit the data better just because it bends more easily. Using the first years to forecast later years checks whether the function captures the real process rather than noise.
Why would young players improve slowly at first?
The authors suggest that learning opportunities and the ability to use them, such as playing more games, studying chess media or regulating practice, grow over childhood, producing an S-shaped curve that first accelerates and later slows.
Does this mean the power law of practice is wrong?
Not for short laboratory tasks. It suggests that curves from brief lab learning do not simply scale up to years-long expertise, where development and changing environments also matter.
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